Projective Representations Research Articles

The presence of the past: analysis & representation of the Strada Novissima

This article aims to take an in-depth look at the first Venice Architecture Biennale of 1980, entitled “The Presence of the Past”, from a contemporary perspective. That means from two points of view: one is broad and testifies to the moment of conflict between an exhausted modern movement and new - and varied - positions that attempt to explore several disciplinary arguments, of which the exhibition is at least a partial reflection; the other is intrinsic to the exhibition and has the Strada Novissima installation as its most faithful interpreter. Along its twenty façades, the ideas of those architects who, according to the curators, are the most representative of this particular moment in time are expressed in projects - or representation of projects - that walk the line between architecture, sculpture, scenography, and painting.

G-frame generator sets for projective unitary representations

<abstract><p>Frames with special structures play a crucial role in various industrial applications, such as medical imaging, quantum communication, recognition and identification software, and so on. In this paper, we will discuss a more general setting, i.e., a g-frame induced by the projective unitary representation. We show some new results on the dilation property for g-frame generator sets for unitary groups and projective unitary representations. In particular, by using complete wandering operators, several properties of g-frame generators for projective unitary representations have been obtained. Moreover, we explore some characterizations of the g-frame generator dual pairs.</p></abstract>

Classification of irreducible based modules over the complex representation ring of $ S_4 $

<abstract><p>The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $ \mathbb{Z}_+ $-modules of finite rank over such a representation ring is also semisimple. In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring $ r(S_4) $ of the symmetric group $ S_4 $. In total, 16 inequivalent irreducible based modules were obtained. In this process, the MATLAB program was used in order to obtain some representation matrices. Based on such a classification result, we further discuss the categorification of based modules over $ r(S_4) $ by module categories over the complex representation category $ {\rm Rep}(S_4) $ of $ S_4 $ arisen from projective representations of certain subgroups of $ S_4 $.</p></abstract>

Стан підтримки малого та середнього підприємництва на регіональному рівні

The entrepreneurial environment in Ukraine is beginning to emerge under the influence of the shock of the transition period associated with the mass release of workers and employees, the collapse of the financial system, the lack of national currency, and, consequently, the lack of funds for people’s livelihood. The article raises a range of issues related to the dilemma of “state support – self-regulation” of entrepreneurial structures, which, despite their constant presence on the policy agenda, find separate solutions at different stages of societal development. The paper notes that the differences in regional development have also affected the state of the entrepreneurial environment at the regional level and the set of measures to support small and medium-sized enterprises (SMEs). The article analyzes SME support programs at the level of the Poltava region and highlights their positive and negative aspects. It is concluded that the regional SME support programs’ positive results are due to factors such as linking financial assistance to the number of jobs created and cooperation of the city SME representatives with NGOs, Poltava City Council, and PROMIS project representation in the region. Attention is paid to the activity of the Regional Fund for Entrepreneurship Support, reflecting the specifics of the economy of the Poltava region. The necessity of new approaches to improve the state support of small and medium-sized enterprises, considering the war risks, conditions of post-war recovery, and the limited state budget, was substantiated. In this context, it is recommended to strengthen the capacity of entrepreneurship through the self-organization of entrepreneurs by creating business associations. It is noted that the associations already registered during the study are actively cooperating with state institutions to improve the conditions for their own economic activity. The proposals and conclusions formulated in the article can be helpful for regional public authorities in forming and implementing SME support policy, as well as individual programs to support and stimulate SME development. Keywords: support, small and medium entrepreneurship, public authorities, program, development, interaction.

Geo-Quantitative Analysis of Violent Crime Intervention Strategies in Kaduna State, Nigeria

The complexities and consequences of violent crimes require a robust and effective approach to achieve a sustainable reduction of violence in our society. This paper, therefore, presents an avenue to provide the geo-quantitative analysis of violent occurrences in Kaduna State of Nigeria to determine the pragmatic interventionist strategies for managing violent crime incidence among the people of the study area. Geo-quantitative analysis entails transforming numerical data about criminal activities into a map of proportional circles to depict the distribution pattern of criminal activities and using scree plots and factor loading to show higher-order intervention measures for managing violent crimes. Five Local Government Areas (LGAs) with relatively high levels of violent incidence were purposively selected while snowball sampling method was used to administer questionnaires to 384 respondents. From the findings, the spatial distribution of violent crimes revealed that Birni-Gwari and Kachia LGAs accounted for the highest incidence of kidnapping and village raids respectively while Kajuru LGA is known for the highest cases of cattle rustling and armed robbery. The critical interventions in managing violent crimes are reconciliation efforts to restore order and normalcy, negotiation between aggrieved parties, and adoption of a traditional conflict resolution system with a rotated factor matrix of 0.828, 0.581, and 0.544 respectively. Furthermore, the strongest positive correlation exists between good government policies and equity with fairness in representation and establishment of developmental projects (r = 0.524, and p ˂ 0.001). Therefore, the study recommended establishing conflict mediation centers across the communities. Similarly, it is essential to have a good governance system that will give people their fair share of national resources and adequate policing of communities that could guarantee the reduction of violent incidents to the barest minimum.

Implementation of Radon transformation by rotating 1D interferometer

The use of interferometry to observe objects in three-dimensional space requires a corresponding rank of the system of interferometric base vectors. The paper considers one of the ways to solve such a problem using the rotation of a 1D interferometer at an angle to the rotation axis This, with the exception of special cases of parallelism or perpendicularity of the axes, makes it possible to form a consecutive 3D interferometer. Using the rotation of the Earth, such an interferometer performs Radon transform of the angular structure of the spatial image when observing sources far beyond the size of the interferometer base. For this, the elements of the interferometer should be placed at different latitudes. The obtained analytical expressions show that the two-dimensional representation of one-dimensional projections as a function of the rotation angle then takes the form of a sinogram. A rotating 1D interferometer can be used in a number of fields of science and technology, for example, to solve location problems, in ultrasonic de¬fectoscopy, in technical vision systems, in radio astronomy, etc.

PRINCIPAL COMPONENTS VERSUS AUTOENCODERS FOR DIMENSIONALITY REDUCTION: A CASE OF SUPER-RESOLVED OUTPUTS FROM PRISMA HYPERSPECTRAL MISSION DATA

Abstract. This study attempts to solve these issues associated with hyperspectral (HS) data, i.e., coarse spatial resolution and high volume, by understanding the effect of deep learning and traditional dimensionality reduction on super-resolved products generated from the recently launched PRecursore IperSpettrale della Missione Applicativa (PRISMA) HS mission. Four single-frame super-resolution (SR) algorithms have been used to super-resolve a 30 m PRISMA scene of Ahmedabad, India and generate 15 m spatial resolution images with both spatial and spectral fidelity. Iterative back projection (IBP) and sparse representation (SIS) are the best and worst-performing SR algorithms following a comparative assessment and validation protocol. Next, denoising autoencoders and PCT computed using singular and eigenvalue decompositions have been executed on the original PRISMA, IBP and SIS-based super-resolved datasets. The resulting low-dimensional representations have been assessed to preserve the original dataset's topology using label-independent Lee and Verleysen's co-ranking matrix and loss of quality measure. Findings suggest that autoencoders are computationally expensive and require a higher neighbourhood size than PCT and its variants to produce a high-quality encoding. These insights remain significant for urban information extraction as there are few direct comparative assessments between machine learning-based linear and non-linear data compression methods in earlier studies.

PDRLRR: A novel low-rank representation with projection distance regularization via manifold optimization for clustering

The low-rank representation (LRR) method has attracted widespread attention due to its excellent performance in pattern recognition and machine learning. LRR-based variants have been proposed to solve the three existing problems in LRR: (1) the projection matrix is permanently fixed when dimensionality reduction techniques are adopted; (2) LRR fails to capture the local geometric structure; and (3) the solution deviates from the real low-rank solution. To address these problems, this paper proposes a low-rank representation with projection distance regularization (PDRLRR) via manifold optimization for clustering. In detail, we first introduce a low-dimensional projection matrix and a projection distance regularization term to fit the projected data automatically and capture the local structure of the data, respectively. Consequently, the projection matrix and representation matrix are obtained jointly. Then, we obtain a more accurate low-rank solution by minimizing the Schatten-p norm instead of the nuclear norm. Next, the projection matrix is optimized through a generalized Stiefel manifold. Extensive experiments demonstrate that our proposed method outperforms the state-of-the-art methods.

On projective Anosov subgroups of symplectic groups

AbstractWe prove that a word hyperbolic group whose Gromov boundary properly contains a 2‐sphere cannot admit a projective Anosov representation into , . We also prove that a word hyperbolic group that admits a projective Anosov representation into is virtually a free group or virtually a surface group, a result established independently by Dey–Greenberg–Riestenberg.

Galilean relativity and the path integral formalism in quantum mechanics

Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.

Graph-Based Keyphrase Extraction for Software Traceability in Source Code and Documentation Mapping

Natural Language Processing (NLP) forms the basis of several computational tasks. However, when applied to the software system’s, NLP provides several irrelevant features and the noise gets mixed up while extracting features. As the scale of software system’s increases, different metrics are needed to assess these systems. Diagrammatic and visual representation of the SE projects code forms an essential component of Source Code Analysis (SCA). These SE projects cannot be analyzed by traditional source code analysis methods nor can they be analyzed by traditional diagrammatic representation. Hence, there is a need to modify the traditional approaches in lieu of changing environments to reduce learning gap for the developers and traceability engineers. The traditional approaches fall short in addressing specific metrics in terms of document similarity and graph dependency approaches. In terms of source code analysis, the graph dependency graph can be used for finding the relevant key-terms and keyphrases as they occur not just intra-document but also inter-document. In this work, a similarity measure based on context is proposed which can be employed to find a traceability link between the source code metrics and API documents present in a package. Probabilistic graph-based keyphrase extraction approach is used for searching across the different project files.

Mapping class group representations and Morita classes of algebras

A modular fusion category \mathcal{C} allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then \mathcal{C} has a unique Morita class of simple non-degenerate algebras, namely, that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.

Predicting Drug-Disease Associations Through Similarity Network Fusion and Multi-View Feature Projection Representation.

Predicting drug-disease associations (DDAs) through computational methods has become a prevalent trend in drug development because of their high efficiency and low cost. Existing methods usually focus on constructing heterogeneous networks by collecting multiple data resources to improve prediction ability. However, potential association possibilities of numerous unconfirmed drug-related or disease-related pairs are not sufficiently considered. In this article, we propose a novel computational model to predict new DDAs. First, a heterogeneous network is constructed, including four types of nodes (drugs, targets, cell lines, diseases) and three types of edges (associations, association scores, similarities). Second, an updating and merging-based similarity network fusion method, termed UM-SF, is presented to fuse various similarity networks with diverse weights. Finally, an intermediate layer-mediated multi-view feature projection representation method, termed IM-FP, is proposed to calculate the predicted DDA scores. This method uses multiple association scores to construct multi-view drug features, then projects them into disease space through the intermediate layer, where an intermediate layer similarity constraint is designed to learn the projection matrices. Results of comparative experiments reveal the effectiveness of our innovations. Comparisons with other state-of-the-art models by the 10-fold cross-validation experiment indicate our model's advantage on AUROC and AUPR metrics. Moreover, our proposed model successfully predicted 107 novel high-ranked DDAs.

Mackey's obstruction map for discrete graded algebras

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles.Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules.The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules.The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields.

Generalized positive energy representations of groups of jets

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra \mathfrak{k} . Consider the Fréchet–Lie group G := J_0^\infty(V; K) of \infty -jets at 0\in V of smooth maps V \to K , with Lie algebra \mathfrak{g}=J_0^\infty(V; \mathfrak{k}) . Let P be a Lie group and write \mathfrak{p}:=\operatorname{Lie}(P) . Let \alpha be a smooth P -action on G . We study smooth projective unitary representations \bar{\rho} of G\rtimes_\alpha P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by \bar{\rho}(G) . We show that this condition imposes severe restrictions on the derived representation d\bar{\rho} of \mathfrak{g}\rtimes \mathfrak{p} , leading in particular to sufficient conditions for {\bar{\rho}}|_{G} to factor through J_0^2(V; K) , or even through K .

Groups with three projective characters

Working over the field of complex numbers, the inequivalent irreducible projective representations of a finite group G with 2-cocycle α are considered. The greatest common divisor of the degrees of the α-characters of these representations is used to obtain information about the degrees of the irreducible αS -characters of a Sylow subgroup S of G. Suppose now that G has exactly three irreducible α-characters. Then the potential form that the three degrees can take is found if G has a nontrivial cyclic Sylow subgroup. However, the main result is to show that G is solvable when the degrees are of the form a, ra and ka with certain restrictions on r and k, most notably that r = 1 or k. This involves detailed analysis of the possible actions of G on the irreducible αN -characters of a normal subgroup N of G.

The EU as an ATM? Media Perception Analysis of Next Generation Funds in Spain

The Covid-19 pandemic has been a turning point in terms of communication and economics within the borders of the EU. Hence, the economic response to the consequences of the pandemic has been different from previous crises. Both factors influence the media’s representation of the European project, and the construction of this image is particularly relevant to generating a favourable public opinion towards the European project. This research aims to determine how the Spanish media represent the Next Generation recovery funds and to determine the main discourses around this issue. We analysed news items disseminated by a sample of six leading Spanish news media through qualitative and quantitative methods by applying content and critical discourse analysis. The selection collects data via Twitter from July 2021 to March 2022. We found that media discourse reflects a pro-European sentiment, departing from previous Eurosceptic views. Next Generation funds have positively influenced Spanish perception of the EU and shifted the narrative towards Europeanisation. The EU’s support for Spanish funds management advances European integration, but concerns about transparency and control remain. The findings show how the Spanish media present a pro-European view, placing the economic response as a window of opportunity for profound political, societal, and economic structural changes in Spain.

Hausdorff dimension of limit sets for projective Anosov representations

Hausdorff dimension of limit sets for projective Anosov representations

Ʌ-Hypernuclear States as Dihadronic Molecules

The study of exotic hypernuclei attracts a great deal of interest in nuclear physics. The reality of heavy hyperon hypernuclei is the subject of intense concern among theoreticians and experimenters in recent years. The core-hyperon model uses to explain abnormal nuclei spectra, recent observations of new exotic heavy hyperon hypernuclei cannot be explained or predicted by ordinary heavy core nuclei. These exotic hypernuclei states are a two-cluster bound states. We calculate the mass spectrum and constituent mass of particles in hypernuclei using the relativistic Schrödinger equation with molecular pseudoharmonic-type potential between particles inside the core and hyperon. Such calculations represent the interaction between the hyperon and the nuclei core. I review recent theoretical studies on the ground states and the excited states of hypernuclei bound states. Finally, we present explicit predictions of the exotic bound states based on the interactions obtained from quantum field theory and the projective unitary representation model. Studies have shown that by increasing the mass number of hyperon-core states, the value of the constituent mass and energy eigenvalue of Ʌ-hypernucleus increases. Also, by growing and increasing the proton number in the (Ʌ-N) states the value of the constituent mass of Ʌ-hyperon increases.

Matrix representations of multivalued projections and least squares problems

Multivalued projections are applied to the study of weighted least squares solutions of linear relations equations (or inclusions) and some of its applications. To this end a matrix representation of multivalued projections with respect to the closure of their ranges is described.